For internal regret minimization, we primarily use an extension of Blackwell's regret minimization given by [1]} ([1]}) also known as regret matching. Regret matching also selects its policy with probability “matching” its past regrets of not switching to that action in the past, but it differs in that it additionally uses a fixed inertia parameter \(\alpha \)  and thus always retains a positive probability of staying in place, with probability approaching 1 as the overall regrets vanishNote that our formulation is slightly different from the formulation given in their original paper in that it allows us to use a very low inertia constant (\(\alpha = 10^{-10}\) ) and thus prevents the procedure from repeating actions until the regrets become very low. We empirically observe that this dramatically accelerates convergence to CE in our experiments with large random games.. Formally, we have
\(\Pr (a^{t+1}_i = a_i) = \left.{\left\lbrace \begin{array}{ll}\frac{\alpha }{\alpha + \sum _{a^{\prime }_i \in A_i} R^{I,t}_{i,+}(a^t_i, a^{\prime }_i)}, & \text{if } a_i = a^t_i \\[10pt]\frac{R^{I,t}_{i,+}(a^t_i, a_i)}{\alpha + \sum _{a^{\prime }_i \in A_i} R^{I,t}_{i,+}(a^t_i, a^{\prime }_i)}, & \text{otherwise}\end{array}\right.}\right\rbrace \) 
